Question: Multiply the following complex numbers, marked as blue dots on the graph: $(5 e^{\pi i / 3}) \cdot ( e^{19\pi i / 12})$ (Your current answer will be plotted in orange.)
Solution: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $5 e^{\pi i / 3}$ ) has angle $\frac{1}{3}\pi$ and radius $5$ The second number ( $ e^{19\pi i / 12}$ ) has angle $\frac{19}{12}\pi$ and radius $1$ The radius of the result will be $5 \cdot 1$ , which is $5$ The angle of the result is $\frac{1}{3}\pi + \frac{19}{12}\pi = \frac{23}{12}\pi$ The radius of the result is $5$ and the angle of the result is $\frac{23}{12}\pi$.